6.5 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Blackman's theorem | 1/1 | https://en.wikipedia.org/wiki/Blackman's_theorem | reference | science, encyclopedia | 2026-05-05T11:45:53.294133+00:00 | kb-cron |
Blackman's theorem is a general procedure for calculating the change in an impedance due to feedback in a circuit. It was published by Ralph Beebe Blackman in 1943, was connected to signal-flow analysis by John Choma, and was made popular in the extra element theorem by R. D. Middlebrook and the asymptotic gain model of Solomon Rosenstark. Blackman's approach leads to the formula for the impedance Z between two selected terminals of a negative feedback amplifier as Blackman's formula:
Z
=
Z
D
1
+
T
S
C
1
+
T
O
C
,
{\displaystyle Z=Z_{D}{\frac {1+T_{SC}}{1+T_{OC}}}\ ,}
where ZD = impedance with the feedback disabled, TSC = loop transmission with a small-signal short across the selected terminal pair, and TOC = loop transmission with an open circuit across the terminal pair. The loop transmission also is referred to as the return ratio. Blackman's formula can be compared with Middlebrook's result for the input impedance Zin of a circuit based upon the extra-element theorem:
Z
i
n
=
Z
i
n
∞
[
1
+
Z
e
0
/
Z
1
+
Z
e
∞
/
Z
]
{\displaystyle Z_{in}=Z_{in}^{\infty }\left[{\frac {1+Z_{e}^{0}/Z}{1+Z_{e}^{\infty }/Z}}\right]}
where:
Z
{\displaystyle Z\ }
is the impedance of the extra element;
Z
i
n
∞
{\displaystyle Z_{in}^{\infty }}
is the input impedance with
Z
{\displaystyle Z\ }
removed (or made infinite);
Z
e
0
{\displaystyle Z_{e}^{0}}
is the impedance seen by the extra element
Z
{\displaystyle Z\ }
with the input shorted (or made zero);
Z
e
∞
{\displaystyle Z_{e}^{\infty }}
is the impedance seen by the extra element
Z
{\displaystyle Z\ }
with the input open (or made infinite). Blackman's formula also can be compared with Choma's signal-flow result:
Z
S
S
=
Z
S
0
[
1
+
T
I
1
+
T
Z
]
,
{\displaystyle Z_{SS}=Z_{S0}\left[{\frac {1+T_{I}}{1+T_{Z}}}\right]\ ,}
where
Z
S
0
{\displaystyle Z_{S0}\ }
is the value of
Z
S
S
{\displaystyle Z_{SS}\ }
under the condition that a selected parameter P is set to zero, return ratio
T
Z
{\displaystyle T_{Z}\ }
is evaluated with zero excitation and
T
I
{\displaystyle T_{I}\ }
is
T
Z
{\displaystyle T_{Z}\ }
for the case of short-circuited source resistance. As with the extra-element result, differences are in the perspective leading to the formula.
== See also == Mason's gain formula
== Further reading == Eugene Paperno (September 2012). "Extending Blackman's formula to feedback networks with multiple dependent sources" (PDF). IEEE Transactions on Circuits and Systems II: Express Briefs. 59 (10): 658–662. Bibcode:2012ITCSE..59..658P. CiteSeerX 10.1.1.695.4656. doi:10.1109/TCSII.2012.2213355. S2CID 8760900. Rahul Sarpeshkar (2010). "§10.7 Driving-point transistor impedances with Blackman's formula". Ultra Low Power Bioelectronics: Fundamentals, Biomedical Applications, and Bio-Inspired Systems. Cambridge University Press. pp. 258 ff. ISBN 9781139485234. Amaldo D'Amico; Christian Falconi; Gianluca Giustolisi; Gaetano Palumbo (April 2007). "Resistance of feedback amplifiers: A novel representation" (PDF). IEEE Transactions on Circuits and Systems II: Express Briefs. 54 (4): 298. Bibcode:2007ITCSE..54..298D. doi:10.1109/TCSII.2006.889713.
== References ==